2 SPECTRUM OF AN ELEMENT OF A NORMED ALGEBRA 317 (e) Let / be an analytic function on a neighborhood of Sp(*), and g an analytic function on a neighborhood of Sp(/(*)). Show that, by restricting the open set on which/is defined, if necessary, the function h = g o/is defined, and that h(x) = g(f(x)). (f) In particular, if Sp(x) is contained in the disk |£| < 1, then for each a e Rthe series converges in A to an element y which commutes with x, and SpQO is the image of Sp(.x) under the mapping £H->(! +£)« (with the determination (1 +£)«=! when £ = 0). In particular, if a = 1/m where m is an integer > 0, then ym = 1 -f x, and we write 12. With the same hypotheses as in Problem 11, let (Kj)1<y<n be a partition of Sp(#) into compact sets. For each;, let U/ be a neighborhood of K,, such that the U/ are n pairwise disjoint. Let// be the analytic function on (J Ufc which takes the value 1 on *=i U/ and 0 on Ufc for k^j. Show that ej = //Jc) is idempotent in A, that e}ek = 0 n whenever A: ^=y, and that V e/ — e . /-i (b) Let A be an isolated point of Sp(*), and let be the Laurent expansion of (& — x)'1 ina neighborhood of A. If eA is the idempotent corresponding to {A} in the partition of Sp(x) consisting of {A} and Sp(jt) — {A}, then e^Xe - x)k = (-l)^.^! for all k ;> 0. (c) Let B be the subalgebra of A generated by e and x. In order that Sp(jc) should consist of only a finite number of points, which are poles of (£e — x)"1, it is necessary r and sufficient that B should be finite-dimensional. (If (£e — x)~l = V F/(£)fl/» where j=i the F, are scalar rational functions, then all powers of x are linear combinations of the aj . Conversely, if B is finite-dimensional, there exists a polynomial/, not identically zero, such that/(x) = 0. Then Sp(x) is contained in the set of zeros of/; if A is a zero of/, of order/?, show also that (Xe — x)pe^ = 0.) (d) Let U be an open neighborhood of Sp(x), having only finitely many connected components U, , and let / be an analytic function on U. Show that f(x) = 0 if and only if the following conditions are satisfied: (1) for each index j such that Sp(jc) n Uj is either infinite or else contains an essential singularity of (£e — x)"1, we have f\Uj — 0; (2) every pole of order p of (& — x)""1 is a zero of/, of order ^p. (For the necessity of (2), argue as in (c).) 13. Let A be a Banach algebra with unit element, and U an open set in C. Show that the set D of elements x e A such that SpA(x) c U is open in A. If / is analytic on U, then the mapping x\— *f(x) of O into A is indefinitely differentiate. 14. Let A be a Banach algebra with unit element denoted by 1 . (a) For each pair jc, y of elements of A and each £ e C, show that p(e^xye"tix) « p(y).if and only if /(£) ^ 0 on Sp(#).n particular, the image of C under the